Analysis Of Incidence Rates

Cover......Page 1
Half Title......Page 2
Series Page
......Page 3
Title Page......Page 4
Copyright Page......Page 5
Table of Contents......Page 6
Preface......Page 16
Author
......Page 18
1.1 Karl Pearson and Spurious Correlation......Page 20
1.2 Jerzy Neyman, Storks, and Babies......Page 22
1.3 Is Poisson Regression the Solution to the Stork Problem?......Page 25
1.4 Further Reading......Page 26
2.2 Closed and Open Populations......Page 28
2.4 Numerators for Rates: Counts......Page 29
2.6 Prevalence Proportions......Page 31
2.7 Denominators for Rates: Count Denominators for Incidence Proportions (Risks)......Page 32
2.8 Denominators for Rates: Person-Time for Incidence Rates......Page 34
2.10 Rate Denominators Other than Person-Time......Page 36
2.11 Different Incidence Rates Tell Different Stories......Page 37
2.12 Potential Advantages of Incidence Rates Compared With Incidence Proportions (Risks)......Page 38
2.13 Potential Advantages of Incidence Proportions (Risks) Compared with Incidence Rates......Page 41
2.15 Radioactive Decay: An Example of Exponential Decline......Page 42
2.17 Relationships Between Rates, Risks, and Hazards......Page 45
2.18 Further Reading......Page 48
3.1 Estimated Associations and Causal Effects......Page 50
3.2 Sources of Bias in Estimates of Causal Effect......Page 51
3.4 Ratios and Differences for Risks and Rates......Page 52
3.5 Relationships between Measures of Association in a Closed Population......Page 53
3.6 The Hypothetical TEXCO Study......Page 54
3.7 Breaking the Rules: Army Data for Companies A and B......Page 58
3.8 Relationships between Odds Ratios, Risk Ratios, and Rate Ratios in Case-Control Studies......Page 60
3.10 Convergence Problems for Estimating Associations......Page 64
3.11 Some History Regarding the Choice between Ratios and Differences......Page 65
3.12 Other Influences on the Choice between Use of Ratios or Differences......Page 67
3.13 The Data May Sometimes Be Used to Choose between a Ratio or a Difference......Page 68
4.1 Alpha Particle Radiation......Page 72
4.3 Prussian Soldiers Kicked to Death by Horses......Page 75
4.4 Variances, Standard Deviations, and Standard Errors for Counts and Rates......Page 76
4.5 An Example: Mortality from Alzheimer’s Disease......Page 80
4.6 Large Sample P-values for Counts, Rates, and Their Differences using the Wald Statistic......Page 81
4.8 Large Sample P-values for Counts, Rates, and Their Differences using the Score Statistic......Page 83
4.9 Large Sample Confidence Intervals for Counts, Rates, and Their Differences......Page 84
4.10 Large Sample P-values for Counts, Rates, and Their Ratios......Page 86
4.12 A Constant Rate Based on More Person-Time Is More Precise......Page 88
4.13 Exact Methods......Page 90
4.14 What Is a Poisson Process?......Page 93
4.15 Simulated Examples......Page 95
4.16 What If the Data Are Not from a Poisson Process? Part 1, Overdispersion......Page 96
4.18 Must Anything Be Rare?......Page 98
4.19 Bicyclist Deaths in 2010 and 2011......Page 100
5.1 Florence Nightingale, William Farr, and Hospital Mortality Rates. Debate in 1864......Page 102
5.2 Florence Nightingale, William Farr, and Hospital Mortality Rates. Debate in 1996–1997......Page 106
5.3 Criticism of Rates in the British Medical Journal in 1995......Page 107
5.4 Criticism of Incidence Rates in 2009......Page 109
6. Stratified Analysis: Standardized Rates
......Page 112
6.2 External Weights from a Standard Population: Direct Standardization
......Page 113
6.3 Comparing Directly Standardized Rates......Page 118
6.4 Choice of the Standard Influences the Comparison of Standardized Rates......Page 121
6.5 Standardized Comparisons versus Adjusted Comparisons from Variance-Minimizing Methods......Page 122
6.7 Variations on Directly Standardized Rates......Page 124
6.8 Internal Weights from a Population: Indirect Standardization......Page 125
6.9 The Standardized Mortality Ratio (SMR)......Page 126
6.10 Advantages of SMRs Compared with SRRs (Ratios of Directly Standardized Rates)......Page 127
6.11 Disadvantages of SMRs Compared with SRRs (Ratios of Directly Standardized Rates)......Page 128
6.13 P-values for Directly Standardized Rates......Page 130
6.14 Confidence Intervals for Directly Standardized Rates......Page 131
6.15 P-values and CIs for SRRs (Ratios of Directly Standardized Rates)......Page 132
6.17 Small Sample P-values and CIs for SMRs......Page 133
6.19 Standardization Is Not Always the Best Choice......Page 134
7.1 Inverse-variance Methods......Page 136
7.2 Inverse-Variance Analysis of Rate Ratios......Page 137
7.3 Inverse-Variance Analysis of Rate Differences......Page 140
7.5 Mantel-Haenszel Methods......Page 141
7.6 Mantel-Haenszel Analysis of Rate Ratios......Page 142
7.7 Mantel-Haenszel Analysis of Rate Differences......Page 143
7.8 P-values for Stratified Rate Ratios or Differences......Page 144
7.9 Analysis of Sparse Data......Page 145
7.11 Stratified Methods versus Regression......Page 146
8.1 What Is Collapsibility?......Page 148
8.2 The British X-Trial: Introducing Variation in Risk......Page 149
8.3 Rate Ratios and Differences Are Noncollapsible because Exposure Influences Person-Time......Page 150
8.4 Which Estimate of the Rate Ratio Should We Prefer?......Page 151
8.5 Behavior of Risk Ratios and Differences......Page 152
8.7 Comparing Risks with Other Outcome Measures......Page 153
8.9 The American X-Cohort Study: 3-Levels of Risk in a Cohort Study......Page 154
8.10 The Swedish X-Cohort Study: A Collapsible Risk Ratio in Confounded Data......Page 158
8.11 A Summary of Findings......Page 160
8.13 Practical Implications: Avoid Common Outcomes......Page 161
8.16 Practical Implications: Uniform Risk......Page 162
8.17 Practical Implications: Use All Events......Page 163
9.1 The Poisson Regression Model for Rate Ratios......Page 164
9.2 A Short Comparison with Ordinary Linear Regression......Page 167
9.3 A Poisson Model without Variables......Page 168
9.4 A Poisson Regression Model with One Explanatory Variable......Page 171
9.6 The Header Information above the Table of Estimates......Page 175
9.7 Using a Generalized Linear Model to Estimate Rate Ratios......Page 177
9.8 A Regression Example: Studying Rates over Time......Page 180
9.9 An Alternative Parameterization for Poisson Models: A Regression Trick......Page 185
9.11 A Short Summary......Page 190
10.1 A Regression Model for Rate Differences......Page 192
10.2 Florida and Alaska Cancer Mortality: Regression Models that Fail......Page 193
10.3 Florida and Alaska Cancer Mortality: Regression Models that Succeed......Page 194
10.5 A Caution......Page 198
11.1 Limitations of Ordinary Least Squares Linear Regression......Page 200
11.2 Florida and Alaska Cancer Mortality Rates......Page 201
11.3 Weighted Least Squares Linear Regression......Page 202
11.4 Importance Weights for Weighted Least Squares Linear Regression......Page 204
11.5 Comparison of Poisson, Weighted Least Squares, and Ordinary Least Squares Regression......Page 205
11.6 Exposure to aCarcinogen: Ordinary Linear Regression Ignores the Precision of Each Rate......Page 210
11.7 Differences in Homicide Rates: Simple Averages versus Population-Weighted Averages......Page 212
11.9 Variance Weighted Least Squares Regression......Page 215
11.10 Cautions regarding Inverse-Variance Weights......Page 218
11.11 Why Use Variance Weighted Least Squares?......Page 219
11.12 A Short Comparison of Weighted Poisson Regression, Variance Weighted
Least Squares, and Weighted Linear Regression......Page 220
11.14 Ratios and Spurious Correlation......Page 223
11.15 Linear Regression with ln (Rate) as the Outcome......Page 226
11.17 Summary......Page 227
12.1 Tabular and Graphic Displays......Page 230
12.2 Goodness of Fit Tests: Deviance and Pearson Statistics......Page 231
12.3 A Conditional Moment Chi-Squared Test of Fit......Page 234
12.4 Limitations of Goodness-of-Fit Statistics......Page 235
12.6 Robust Variance Estimator as a Test of Fit
......Page 236
12.8 Comparing Models using Akaike and Bayesian Information Criterion......Page 237
12.9 Example 1: Using Stata’s Generalized Linear Model Command to Decide
between a Rate Ratio or a Rate Difference Model for the Randomized
Controlled Trial of Exercise and Falls......Page 239
12.10 Example 2: A Rate Ratio or a Rate Difference Model for Hypothetical Data
Regarding the Association between Fall Rates and Age......Page 243
12.11 A Test of the Model Link......Page 246
12.14 A Caution......Page 249
12.15 Further Reading......Page 250
13.1 Estimating the Variance without Regression......Page 252
13.2 Poisson Regression......Page 253
13.3 Rescaling the Variance using the Pearson Dispersion Statistic......Page 254
13.4 Robust Variance......Page 257
13.6 Using the Robust Variance to Study Length of Hospital Stay......Page 258
13.8 The Bootstrap Idea......Page 261
13.10 The Bootstrap Percentile Method......Page 262
13.13 The Bootstrap-T Method
......Page 263
13.15 Permutation and Randomization......Page 264
13.17 Better Randomization Using the Randomized Block Design of the Original Study......Page 265
13.18 A Summary......Page 266
14.1 Neyman’s Approach to His Data......Page 268
14.2 Using Methods for Incidence Rates......Page 269
14.3 A Model That uses the Stork/Women Ratio......Page 271
15.2 Quadratic Splines......Page 274
15.3 Fractional Polynomials......Page 276
15.5 Which Method Is Best?......Page 278
16.1 An Example: Shoes and Falls......Page 280
16.2 Problem 1: Using Subgroup P-values for Interpretation......Page 282
16.3 Problem 2: Failure to Include Main Effect Terms When Interaction Terms Are Used......Page 283
16.5 Problem 4: Interaction May Be Present on a Ratio Scale but Not on a
Difference Scale, and Vice Versa......Page 284
16.6 Problem 5: Failure to Report All Subgroup Estimates in an Evenhanded Manner......Page 285
17. Negative Binomial Regression
......Page 290
17.2 An Example: Accidents among Workers in a Munitions Factory......Page 292
17.3 Introducing Equal Person-Time in the Homicide Data......Page 296
17.4 Letting Person-Time Vary in the Homicide Data......Page 298
17.5 Estimating a Rate Ratio for the Homicide Data......Page 300
17.6 Another Example using Hypothetical Data for Five Regions......Page 302
17.7 Unobserved Heterogeneity......Page 305
17.8 Observing Heterogeneity in the Shoe Data......Page 306
17.10 A Rate Difference Negative Binomial Regression Model......Page 307
17.11 Conclusion......Page 310
18. Clustered Data
......Page 312
18.1 Data from 24 Fictitious Nursing Homes......Page 313
18.3 A Single Random Set of Data for the Nursing Homes......Page 314
18.4 Variance Adjustment Methods......Page 318
18.5 Generalized Estimating Equations (GEE)......Page 320
18.6 Mixed Model Methods......Page 322
18.7 What Do Mixed Models Estimate?......Page 323
18.9 Simulation Results for Some Mixed Models......Page 324
18.11 Which Should We Prefer for Clustered Data, Variance-Adjusted or Mixed Models?......Page 327
18.13 Further Reading......Page 328
19.2 Using Rates to Evaluate Governmental Policies......Page 330
19.3 Study Designs for Governmental Policies......Page 331
19.4 A Fictitious Water Treatment and U.S. Mortality 1999–2013......Page 332
19.5 Poisson Regression......Page 333
19.6 Population-Averaged Estimates (GEE)......Page 335
19.7 Conditional Poisson Regression, a Fixed-Effects Approach......Page 336
19.8 Negative Binomial Regression......Page 339
19.10 Water Treatment in Only 10 States......Page 340
19.12 A Published Study......Page 343
20.2 Matching in Randomized Controlled Trials......Page 346
20.3 Matching in Cohort Studies......Page 347
20.4 Matching to Control Confounding in Some Randomized Trials and Cohort Studies......Page 348
20.5 A Benefit of Matching; Only Matched Sets with at Least One Outcome
Are Needed......Page 349
20.6 Studies Designs that Match a Person to Themselves......Page 351
20.7 A Matched Analysis Can Account for Matching Ratios that Are Not Constant......Page 352
20.9 Stratified Methods for Estimating Risk Ratios for Matched Data......Page 353
20.10 Odds Ratios, Risk Ratios, Cell A, and Matched Data......Page 358
20.11 Regression Analysis of Matched Data for the Odds Ratio......Page 360
20.12 Regression Analysis of Matched Data for the Risk Ratio......Page 363
20.13 Matched Analysis of Rates with One Outcome Event......Page 368
20.14 Matched Analysis of Rates for Recurrent Events......Page 374
20.15 The Randomized Trial of Exercise and Falls; Additional Analyses......Page 378
20.16 Final Words......Page 379
21.1 What Are Margins?......Page 380
21.3 Estimating a Rate Difference from a Rate Ratio Model......Page 381
21.4 Death by Age and Sex: A Short Example......Page 382
21.5 Skunk Bite Data: A Long Example......Page 385
21.6 Obtaining the Rate Difference: Crude Rates......Page 386
21.7 Using the Robust Variance......Page 389
21.8 Adjusting for Age......Page 390
21.9 Full Adjustment for Age and Sex......Page 395
21.10 Marginal Commands for Interactions......Page 397
21.11 Marginal Methods for a Continuous Variable......Page 401
21.12 Using a Rate Difference Model to Estimate a Rate Ratio: Use the ln Scale......Page 403
22.1 Cancer Mortality Rate in Alaska......Page 406
22.2 The Rate Ratio for Falling in a Trial of Exercise......Page 410
23.1 A Simple Example......Page 414
23.2 A Perfectly Predicted Outcome......Page 416
23.3 Memory Problems......Page 417
23.4 A Caveat......Page 420
24.2 Analysis by Treatment Received May Yield Biased Estimates of Treatment Effect......Page 422
24.3 Using an Instrumental Variable......Page 424
24.4 Two-Stage Linear Regression for Instrumental Variables......Page 427
24.5 Generalized Method of Moments......Page 428
24.6 Generalized Method of Moments for Rates......Page 430
24.8 There Is No Free Lunch......Page 433
24.9 Final Comments......Page 434
25.2 Poisson Regression and Exponential Proportional Hazards Regression......Page 436
25.3 Poisson and Cox Proportional Hazards Regression......Page 440
25.4 Hypothetical Data for a Rate that Changes over Time......Page 442
25.5 A Piecewise Poisson Model......Page 445
25.6 A More Flexible Poisson Model: Quadratic Splines......Page 446
25.7 Another Flexible Poisson Model: Restricted Cubic Splines......Page 449
25.8 Flexibility with Fractional Polynomials......Page 450
25.9 When Should a Poisson Model Be Used? Randomized Trial of a
Terrible Treatment......Page 452
25.10 A Real Randomized Trial, the PLCO Screening Trial......Page 453
25.11 What If Events Are Common?......Page 455
25.13 Collapsibility and Survival Functions......Page 456
25.14 Relaxing the Assumption of Proportional Hazards in the Cox Model......Page 457
25.15 Relaxing the Assumption of Proportional Hazards for the Poisson Model......Page 459
25.16 Relaxing Proportional Hazards for the Royston-Parmar Model......Page 461
25.17 The Life Expectancy Difference or Ratio......Page 462
25.19 A Short Summary
......Page 465
Bibliography......Page 468
Index......Page 486